3. Experimental Methods 43
3.2. Disperse Phase
3.2.1. Inertial-particle Staging
Particles are preloaded into the chamber and become fluidised, forming a continuous vertical particle stream. The particle stream always has a net upward direction in the certerline, where the flow is fastest, but can fall downward where the flow is slower, closer to the walls. Once the particles reach the bottom, they are collected by the funnel insert and they float up again.
Figure3.4shows a cut away of the apparatus base, containing a particle staging platform.
It is composed of fine wire mesh and is secured at the bottom of a funnel-like insert. Air passing from the underside of the insert entrains the particles in the flow, provided that the gas velocity is faster thanut of the particles. The apparatus opperates at pressures at which the terminal velocity may depend upon the pressure. In figure3.16,ut(dp,ρgas)is plotted for three example pressures: 0.5,1.0 and 5.0 mbar. The dashed line is calculated using the Stokes drag law and the solid lines represent the Epstein drag (recall that Stokes drag is proportion tod2pand Epstein drag directly todp). The point at which the three pressure-dependent lines intersect the dashed curve, corresponding to 9/2λ, happens for increasingly large particle diameter at higher pressures (not shown for 0.5 or 1.0 mbar). At 5 mbar, all particles 80µm and smaller are below the Stokes-Epstein division expressed by equation1.9. Comparing the center-line velocity of the gas of 1.4 ms−1, taken from the calculation shown in figure4.2, to ut(dp,ρgas) delineates the rough pressure region at which the gas velocity will be sufficient to suspend the particles; ugas≥ut(dp,ρgas)is between 1 and 5 mbar for steel particles of dp=15−65µm.
Figure 3.16.: Terminal velocity of steel particles,ρp=8050 kg m−3, vs. particle size for three different operational pressures. Dashed lines represent the stokes drag law and solid lines the Epstein drag law.
The pressure at which particles will float and that at which they cannot is bridged by the transition from SS1 to SS2 (see figure3.6). During SS1, the particles sit on the staging platform8. That is, the pressure of SS1 corresponds to too high of a terminal velocity for the particles to float. The point labeled ‘equilibration’ in figure3.6is when the particles first couple to the flow. The particle stream circulates for an indefinite amount of time during SS2.
The particles are able to circulate because the gas velocity decreases closer to the walls, as shown in figure3.15, and so particles that migrate out of the centerline of the flow fall down to the bottom of the container. There, they are collected by the tapered tray and land on the mesh staging platform, where they can be entrained in the flow again.
In practice, I find that the lowest pressure at which I can suspend solid particles is for a SS2 between 2.5 and 3 mbar, consistent with the above considerations.
8They are perturbed and float during the initial evacuation, but they settle back down even before the pressure reaches SS1.
Figure 3.17.:Preparation of a sample of steel particles 15-65µm, weighing a total of 120 mg.
The mass of the gas in the apparatus when the pressure is 1 mbar is 12 milligrams. To target a value ofε around 1, I weighed the particles before putting them onto the staging platform. They are injected while the system is at ambient conditions by putting them into a pipette whose tip is inserted at H1. Blowing lightly compressed air through the pipette blows the particles into the chamber. Because I found that the flow apparently selects a subsection of the particle size distribution (demonstrated in the next chapter), I introduced an amount closer to 120 milligrams. Figure3.17shows 120 mg of steel particles. Although the system is mainly closed, I found that the particle count could diminish9, requiring me to add more particles. For all of these reasons, the bacground value ofε can only be moderately controlled. However, it can be measured easily a posteriori from the particle-tracking data.
In the next chapter I report on the value ofε achived in the experiments.
3.2.2. Particle Containment
Particles that reach the top of the sedimentation vessel, rather than sedimenting along the sides of the walls, enter an expansion unit at the top of the tube (indicated in figure 3.1 and3.2). In the expansion, the tube cross section increases and therefore the gas velocity decreases. There is also a fine metallic mesh screen at the top of the expansion, to prevent any stray particles from continuing upstream to the vacuum pump.
To decide upon the dimensions of the expansion, I updated the velocity calculation of a single particle of typical size and density in our experiments as it passed through the hollow cone, of heightland opening angleγ. The momentum equation for a single sphere with a single velocity componentupz, given by equation1.16, can be solved analytically, if the gas velocity is held constant. Therefore, I first considered a simple force balance between gravity
9Most likely because they lodge in the corners of the apparatus’ joints.
and Stokes drag, with g=-9.8m/s,
which reduces to equation1.11astbecomes large in a static fluid. As the particle travels upwards in the expansion chamber, however, it finds itself at a new value ofugas, given by equation3.2, replacinghwithland witha(l)depending uponγ. In order to use the simple analytic expression of3.6, I conceptually subdivided the cone into horizontal slices, δl, corresponding to discrete time stepsδt. I made the assumption that the gas velocity within a given slice would be constant and therefore the initial conditions of the particle and gas velocities at each time could be given by their values at the previous time step, provided thatδt>Tf. This way, I updatedcat every time step and stopped the calculation whenupz reached zero, returning the value oflfor which this occurred. I variedγ and limitedlbased upon practical constraints, i.e. it would not be possible to weld the cone onto its supportive
base if the angle were greater than∼30◦and the installation would be feasible forlequal to a few×10 cm. Note that the derivation above holds for particles in the Epstein drag regime as well, however a much faster gas flow is required for the same size particles in the Stokes regime.
3.2.3. Particle Tracking
To study the particles’ dynamics, I track their positions in time using the stereoscopic camera arrangement shown in figure3.1and from the positions derive the velocities and accelerations of the particles at all times. The first and second derivatives of the position are calculated using a finite difference scheme, convolved with a gaussian filter. While particle tracking done in water with polystyrene particles, or performed on water droplets might use backscattered light as the illumination source, I use backlighting to image my solid steel particles in shadow, using three 10-Watt light emitting diode (LED) spotlights. The LED beams are expanded using lenses of diameter and focal length of 75mm and 85mm, respectively. A closeup of the spotlight and lens, mounted on optical posts, is shown in figure 3.18.
The centerline gas flow speed of 1.4 m/s measured by the PIV experiments is fast; particles coupled to the flow are expected to travel a significant fraction of this velocity - at least 20-50 cm/s. Therefore I used high-speed phantom v10 cameras at a framerate of 2000 s−1. The cameras were set upon a leveled optical table and I designed custom mounts for them, shown in figure3.19. The camera mounts have translational freedom on three axis and rotational freedom in one plane. Adjustments to the heightcan be made by loosening the optical posts upon which they sit and turning the threaded bolster on the underside of the mounting plate.
The camera appertures are supported from below by a piece of aluminium that extends from the mounting plate. An optional feature is to use brackets to mount the camera sideways, in
case one wants to view the particle stream in landscape10.
10This configuration was used for initial tests using the Phantom 65 camera, in order to exploit the camera’s large chip in viewing the centerline of the flow. The Phantom 65 cameras have slower recording speeds than the Phantom 10, however, so they were not used for the experiments.
Figure 3.18.: 10-Watt LED lamp with beam-expanding lens. Used as backlighting for Lagrangian Particle Tracking.
Figure 3.19.: Camera mounts with translational and rotation freedom. Holding Phantom 10 cameras that are fitted with 200mm Macro Nikon apertures.
The particle tracking algorithm that produced the particle trajectories analyzed in the Experimental Results chapter is a variation on the standard particle tracking procedure [Ouellette 2006 ] consisting of the following steps in the stated order: particle image finding, stereoscopic reconstruction, tracking in time. However, each of these steps has been changed from the originally stated version to fit the specific needs and limitations of the current experimental setup. Below I describe each step in more detail11.
3.2.4. Particle Image Finding
The first step of the tracking algorithm is a precise localization of the particle images on each camera sensor. For well-lit and well-focused particle images produced for example by laser
11The remainder of this section closely follows notes written by Jan Molacek describing the customized camera model that he implemented for this project.
illumination of tracer particles in water, one can choose from a number of methods such as center-of-mass, 1-D gaussian fit, 2-D gaussian fit, or logarithmic fit. The particular choice of method is determined by the relative value placed on speed as opposed to accuracy. In the present setup, due to the comparably weak illumination producing a low signal to noise ratio, and the broad form of the point spread function, only the 2-D gaussian fit method is capable of reliably extracting the particle image positions. Figure3.20shows a single particle in shadow, its intensity map, and slices in the intensity.
0 25 0
20 40
0 25
0 20 40
0 5 10 15 20 25
x (pixels) 100
150
in te nsi ty
6 7 8 9 10 11 12 13 14
Figure 3.20.: Top: shadowgraph of a steel particle. Center: Intensity map of the particle.
Bottom, slices in intensity, the legend shows the y-position of the slice. The spatial resolution is∼12µm/pixel.
First, the raw video is read once in full to determine the median intensity of each pixel, which is considered to be a sufficiently good approximation to the background intensity due to the relatively low particle density used. Next, the video is read once more, but this time the pixel intensity of each frame is subtracted from the median pixel intensity to obtain the processed frame, which should only show the intensity variation due to particle presence, with particle locations corresponding to local intensity maxima. All local intensity maxima above a given threshold (in our setup the threshold value was chosen to be 10) are considered potential approximate locations of particle images. For each such local maximum, a square window of side length 13 pixels and centered on the local maximum is selected as the set of pixels on which the fitting (see below) will be performed. In case that local maximum is less than 7 pixels away from the frame boundary, the square window is appropriately cropped to fit inside the video frame.
I try to fit the pixel intensity profile with the following function of 6 parameters (in its most general form):
f(x) =Aexp
κ4(x−x0)2+κ5(x−x0)(y−y0) +κ6(y−y0)2 , (3.8)
wherex= [x,y]denotes the pixel position on the camera sensor. I assume that f(x)→0 for
|x| →∞, which impliesκ4,κ6<0 andκ4κ6−κ52/4>0.
Let us denote the number of pixels within the fitting set described above asN, the location ofn-th pixel from the fitting set as[xn,yn], and its intensitypn. To simplify the subsequent algebra, I also introduce for each pixel a vectorgn=
1,x¯n,¯yn,x¯2n,x¯ny¯n,y¯2n
with ¯xn=xn−x0,
¯
yn=yn−y0and writegni for thei-th component ofgn. Thus I can write the value of the fitting function at the location ofn-th pixel as
fn=exp
withκ2=κ3=0 (this constraint effectively definesx0andy0).
The best fit is then defined as that combination of parameters κi, x0, and y0 which minimizes the sum
In order to obtain an initial estimate of the fitting parameters that is needed for the subsequent iterative refinement, I use the first two moments of pixel intensities over the fitting set. For an intensity profile given by (3.8), I have
S0≡ parameters is achieved by calculating the following integrals,
S¯0≡
Given a starting guess for the fitting parametersκ=κm(with the initial guessκ0obtained
via the procedure described above), I need a way to refine the guess to further reduce the sum (3.10). Consider introducing a prefactor in (3.10):
N
I want to find thoseciwhich minimise the sum in (3.13). This is achieved by differentiating (3.13) with respect to eachcq, giving
N
which means thatcsolves the system
Mc˜ =R˜ with M˜i j=
Oncecis obtained, an improved guess forκgiven starting guessκmis obtained byκim+1= κim+ci/c1fori6=1 andκ1m+1=κ1m+lnc1, since
The iterative procedure is run repeatedly until either the fitting parameters converge to within the desired precision, the fitting parameters diverge, or the number of iterative steps reaches a set limit. In the last two cases, I discard the results and do not consider the given local intensity maximum as a particle image location.